How to modify mesh after sculpting with multiresolution modifier?
The Multires modifier does not like changes on the low-poly base mesh very much. It can mess up the high-poly levels and add spikes and other abnormalities.
You can try to select and separate the new extruded mesh, remove the Multires modifier from the separated mesh, then join it again with the body object (that has the Multires modifier). Make the body object the active object (selected last) to keep the Multires modifier.
If this doesn't work then it's time for plan B.
Plan B: Shrinkwrap
Make a copy of the original unchanged object, remove the Multires modifier from the copy, add the inside of the mouth to the copy, add the Multires modifier. Finally, add a Shrinkwrap modifier and shrinkwrap it to the original mesh to transfer the sculpted details. For a better result use Wrap Method Project, and tick the Negative checkbox (project direction). Then apply the Shrinkwrap modifier.
Example videos:

Blender Sculpting Tutorial: Full Advanced Creature Creation Workflow
Projecting High-Res Models in Blender 2.8 using Shrinkwrap  (use a Multires modifier not a Subdivision Surface modifier like in this video)